AI 中文总结
本文用群环方法研究 Alexander quandle 环，识别增广幂与交换子子代数，确定有限循环及二面体 quandle 的增广商，并分类幂等元与证明 Peirce 谱猜想。
AI 中文摘要
对于阿贝尔群 $G$ 上带有自同构 $\phi$ 的 Alexander quandle，我们在任意交换幺环系数环上的普通群环中，将其增广幂和非单位交换子子代数识别为显式理想。每个左范数增广幂也是所有具有相同因子数的乘积的生成张成空间，与括号化方式无关。在 $\mathbb{Z}$ 上，我们确定了非平凡有限循环群上 Alexander quandle 的每一个逐次增广商。对于阶为 $m \ge 4$ 的偶数二面体 quandle，次数 $r \ge 2$ 的每个商都是 $(\mathbb{Z}/(m/2)\mathbb{Z})^2$；这证明了针对被推翻的阶为 $m$ 的预测所提出的替代方案。对于 $\operatorname{Core}(\mathbb{Z})$ 的整数 quandle 环，我们描述了所有增广幂，确定了非单位交换子子代数，并证明了由 1 和 0 索引的基元素之差不在增广理想的平方中。当 $(1-\phi)G$ 是有限 $p$-群时，整数滤过是分离的，因此每个非零整数幂等元都具有增广一。傅里叶分析表明，有限奇数阶交换 Alexander quandle 只有零和基元素作为整数幂等元。结合中缀交换 quandle 的中点描述，这给出了每个有限中缀交换 quandle 的相同分类。最后，任何非平凡奇数阶有限阿贝尔群的核心的复数 quandle 代数具有右 Peirce 谱 $\{0,1,-1\}$，证明了猜想中的奇数阶二面体谱。
英文摘要
For an Alexander quandle on an abelian group $G$ with automorphism $ϕ$, we identify its augmentation powers and nonunital commutator subalgebra with explicit ideals in the ordinary group ring over any commutative unital coefficient ring. Each left-normed augmentation power is also the span of all products with the same number of factors, independently of parenthesization. Over $\mathbb{Z}$, we determine every successive augmentation quotient for Alexander quandles on nontrivial finite cyclic groups. For an even dihedral quandle of order $m \ge 4$, every quotient in degree $r \ge 2$ is $(\mathbb{Z}/(m/2)\mathbb{Z})^2$; this proves the proposed replacement for the disproved order-$m$ prediction. For the integral quandle ring of $\operatorname{Core}(\mathbb{Z})$, we describe all augmentation powers, determine the nonunital commutator subalgebra, and prove that the difference of the basis elements indexed by 1 and 0 does not lie in the square of the augmentation ideal. When $(1-ϕ)G$ is a finite $p$-group, the integral filtration is separated, so every nonzero integral idempotent has augmentation one. Fourier analysis shows that finite odd-order commutative Alexander quandles have only zero and the basis elements as integral idempotents. Combined with the midpoint description of medial commutative quandles, this gives the same classification for every finite medial commutative quandle. Finally, the complex quandle algebra of the core of any nontrivial finite abelian group of odd order has right Peirce spectrum $\{0,1,-1\}$, proving the conjectured odd-order dihedral spectrum.
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